Note on rainbow cycles in edge-colored graphs

Let Gbe a graph of order nwith an edge-coloring c, and let delta(c)(G) denote the minimum color-degree of G. A subgraph Fof Gis called rainbow if all edges of Fhave pairwise distinct colors. There have been a lot of results on rainbow cycles of edge-colored graphs. In this paper, we show that (i) if delta(c)(G) > 2n-1/3, then every vertex of Gis contained in a rainbow triangle; (ii) if delta(c)(G) > 2n-1/3and n >= 13, then every vertex of Gis contained in a rainbow C-4; (iii) if G is complete, n >= 7k - 17and delta(c)(G) > n-1/2+ k, then Gcontains a rainbow cycle of length at least k, where k >= 5.